BEARINGS and LUBRICATION.
§ 99. Work of Friction. Moment of Friction.—The work performed in a unit of time in overcoming the friction of a pair of surfaces is the product of the friction by the velocity of sliding of the surfaces over each other, if that is the same throughout the whole extent of the rubbing surfaces. If that velocity is different for different portions of the rubbing surfaces, the velocity of each portion is to be multiplied by the friction of that portion, and the results summed or integrated.
When the relative motion of the rubbing surfaces is one of rotation, the work of friction in a unit of time, for a portion of the rubbing surfaces at a given distance from the axis of rotation, may be found by multiplying together the friction of that portion, its distance from the axis, and the angular velocity. The product of the force of friction by the distance at which it acts from the axis of rotation is called the moment of friction. The total moment of friction of a pair of rotating rubbing surfaces is the sum or integral of the moments of friction of their several portions.
To express this symbolically, let du represent the area of a portion of a pair of rubbing surfaces at a distance r from the axis of their relative rotation; p the intensity of the normal pressure at du per unit of area; and f the coefficient of friction. Then the moment of friction of du is fprdu ;
the total moment of friction is f f pr. du ; (59) and the work performed in a unit of time in overcoming friction, when the angular velocity is a, is of f pr. du.
It is evident that the moment of friction, and the work lost by being performed in overcoming friction, are less in a rotating piece as the bearings are of smaller radius. But a limit is put to the diminution of the radii of journals and pivots by the conditions of durability and of proper lubrication, and also by conditions of strength and stiffness.
§ Too. Total Pressure between Journal and Bearing.—A single
piece rotating with a uniform velocity has four mutually balanced forces applied to it: (I) the effort exerted on it by the piece which drives it; (2) the resistance of the piece which follows it—which may be considered for the purposes of the present question as useful resistance; (3) its weight; and (4) the reaction of its own cylindrical bearings. There are given the following data:—
(53)
R„ being taken to represent useful and Rp prejudicial resistances. The more nearly the efficiency of a machine approaches to unity the better is the machine.
§ 92. Power and Effect.—The power of a machine is the energy exerted, and the effect the useful work performed, in some interval of time of definite length, such as a second, an hour, or a day.
The unit of power, called conventionally a horse-power, is 550 foot-pounds per second, or 33,000 foot-pounds per minute, or 1,980,000 foot-pounds per hour.
§ 93. Modulus of a Machine.—In the investigation of the proper-ties of a machine, the useful resistances to be overcome and the useful work to be performed are usually given. The prejudicial resistances are generally functions of the useful resistances of the weights of the pieces of the mechanism, and of their form and arrangement; and, having been determined, they serve for the computation of the lost work, which, being added to the useful work, gives the expenditure of energy required. The result of this investigation, expressed in the form of an equation between this energy and the useful work, is called by Moseley the modulus of the machine. The general form of the modulus may be expressed thus
E=U+c(U, A) +¢(A), (54) where A denotes some quantity or set of quantities depending on the form, arrangement, weight and other properties of the mechanism. Moseley, however, has pointed out that in most cases this equation takes the much more simple form of
Pds = Pvdt = Pradt = M adt (57)
is the energy exerted by the couple M in the interval dt; and a similar equation gives the work performed in overcoming a resisting couple. When several couples act on one piece, the resultant of their moments is to be multiplied by the common angular velocity of the whole piece.
§ 96. Reduction of Forces to a given Point, and of Couples to the
Axis of a given Piece.—In computations respecting machines it is often convenient to substitute for a force applied to a given point, or a couple applied to a given piece, the equivalent force or couple applied to some other point or piece; that is to say, the force or
The direction of the effort.
The direction of the useful resistance.
The weight of the piece and the direction in which it acts. The magnitude of the useful resistance.
The radius of the bearing r.
The angle of repose ¢, corresponding to the friction of the journal on the bearing.
And there are required the following:
The direction of the reaction of the bearing.
The magnitude of that reaction.
The magnitude of the effort.
Let the useful resistance and the weight of the piece be compounded by the principles of statics into one force, and let this be called the given force.
The directions of the effort and of the given force are either parallel or meet in a point. If they are parallel, the direction of the reaction of the bearing is also parallel to them; if they meet in a point, the direction of the reaction traverses the same point.
Also, let AAA, fig. 128, be a section of the bearing, and C its axis; then the direction of the reaction, at the point where it intersects the circle AAA, must make the angle 4. with the radius of that circle; that is to say, it must be a line such as PT touching the smaller circle BB, whose radius is r. sin rp. The side on which it touches that circle is determined by the fact that the obliquity of the reaction is such as to oppose the rotation.
Thus is determined the direction of the reaction of the bearing; and the magnitude of that reaction and of the effort are then found by the principles of the equilibrium of three forces already stated in § 7.
The work lost in overcoming the friction of the bearing is the same as that which would be performed in overcoming at the circumference of the small circle BB a resistance equal to the whole pressure between the journal and bearing.
In order to diminish that pressure to the smallest possible amount, the effort, and the resultant of the useful resistance, and the weight of the piece (called above the " given force ") ought to be opposed to each other as directly as is practicable consistently with the purposes of the machine.
An investigation of the forces acting on a bearing and journal lubricated by an oil bath will be found in a paper by Osborne Reynolds in the Phil. Trans., pt. i. (1886). (See also BEARINGS.)
§ 1o1. Friction of Pivots and Collars.—When a shaft is acted upon by a force tending to`shift it lengthways, that force must be balanced by the reaction of a bearing against a pivot at the end of the shaft; or, if that be impossible, against one or more collars, or rings projecting from the body of the shaft. The bearing of the pivot is called a step or footstep. Pivots require great hardness, and are usually made of steel. The flat pivot is a cylinder of steel having a plane circular end as a rubbing surface. Let N be the total pressure sustained by a flat pivot of the radius r; if that pressure be uniformly distributed, which is the case when the rubbing surfaces of the pivot and its step are both true planes, the intensity of the pressure is
p = N/irr2 ; (6o) and, introducing this value into equation 59, the moment of friction of the flat pivot is found to be
3fNr (61) or two-thirds of that of a cylindrical journal of the same radius under the same normal pressure.
The friction of a conical pivot exceeds that of a flat pivot of the same radius, and under the same pressure, in the proportion of the side of the cone to the radius of its base.
The moment of friction of a collar is given by the formula—e r2—r"
3fN r2—r'2' (62)
where r is the external and r' the internal radius.
In the cup and ball pivot the end of the shaft and the step present two recesses facing each other, into which are fitted two shallow cups of steel or hard bronze. Between the concave spherical surfaces of those cups is placed a steel ball, being either a complete sphere or a lens having convex surfaces of a somewhat less radius than the concave surfaces of the cups. The moment of friction of this pivot is at first almost inappreciable from the extreme smallness of the radius of the circles of contact of the ball and cups, but, as they wear, that radius and the moment of friction increase.
It appears that the rapidity with which a
rubbing surface wears away is proportional to
the friction and to the velocity jointly, or nearly
so. Hence the pivots already mentioned wear
unequally at different points, and tend to alter their figures. Schiele
has invented a pivot which preserves its original figure by wearing
equally at all points in a direction parallel to its axis. The following are the principles on which this equality of wear depends
:
The rapidity of wear of a surface measured in an oblique direction is to the rapidity of wear measured normally as the secant of the obliquity is to unity. Let OX (fig. 129) be the axis of a pivot, and let RPC be a portion of a curve such that at any point P the secant of the obliquity to the normal of the curve of a line parallel to the axis is inversely proportional to the ordinate PY, to which the velocity of P is proportional. The rotation of that curve round OX will generate the form of pivot required. Now let PT be a tangent to the curve at P, cutting OX in T ; PT = PY X secant obliquity, and this is to be a constant quantity; hence the curve is that known as the tractory of the straight line OX, in which PT -=- OR =constant. This curve is described by having a fixed straight edge parallel to OX, along which slides a slider carrying a pin whose centre is T. On that pin turns an arm, carrying at a point P a tracing-point, pencil or pen. Should the pen have a nib of two jaws, like those of an ordinary drawing-pen, the plane of the jaws must pass through PT. Then, while T is slid along the axis from 0 towards X, P will be drawn after it from R towards C along the tractory. This curve, being an asymptote to its axis, is capable of being indefinitely prolonged towards X; but in designing pivots it should stop before the 'angle PTY becomes less than the angle of repose of the rubbing surfaces, otherwise the pivot will be liable to stick in its bearing. The moment of friction of " Schiele's anti-friction pivot," as it is called, is equal to that of a cylindrical journal of the radius OR=PT the constant tangent, under the same pressure.
Records of experiments on the friction of a pivot bearing will be found in the Proc. Inst. Mech. Eng. (1891), and on the friction of a collar bearing ib. May 1888.
§ 102. Friction of Teeth.—Let N be the normal pressure exerted between a pair of teeth of a pair of wheels; s the total distance through which they slide upon each other; n the number of pairs of teeth which pass the plane of axis in a unit of time; then
of Ns (63) is the work lost in unity of time by the friction of the teeth. The sliding s is composed of two parts, which take place during the
approach and recess respectively. Let those be denoted by si and s2, so that s=si+s2. In § 45 the velocity of sliding at any instant has been given, viz. u=c (ai+a2), where u is that velocity, c the distance TI at any instant from the point of contact of the teeth to the pitch-point, and a1, a2 the respective angular velocities of the wheels.
Let v be the common velocity of the two pitch-circles, ri, r2, their radii; then the above equation becomes
I I
u = cv
ri+r2
To apply this to involute teeth, let ci be the length of the approach, c2 that of the recess, ui, the mean volocity of sliding during the approach, u2 that during the recess; then 1/
n1 2v (1 , --+ r2/ ' u2 2v \ rl+ r2)
also, let B be the obliquity of the action; then the times occupied by the approach and recess are respectively cl Cl
v cos B' v cos B'
giving, finally, for the length of sliding between each pair of teeth, c12+c22 ( I \1'1+r1)
S=Sl+s2=2 cos
(64)
which, substituted in equation (63), gives the work lost in a unit of time by the friction of involute teeth. This result, which is exact for involute teeth, is approximately true for teeth of any figure.
For inside gearing, if r1 be the less radius and r2 the greater,
I _ I is to be substituted for I I
r1 r2 ri r2
§ 103. Friction of Cords and Belts.—A flexible band, such as a cord, rope, belt or strap, may be used either to exert an effort or a resistance upon a pulley round which it wraps. In either case the tangential force, whether effort or resistance, exerted between the band and the pulley is their mutual friction, caused by and proportional to the normal pressure between them.
Let T1 be the tension of the free part of the band at that side towards which it tends to draw the pulley, or from which the pulley tends to draw it; T2 the tension of the free part at the other side; T the tension of the band at any intermediate point of its arc of contact with the pulley; B the ratio of the length of that arc to the radius of the pulley; dB the ratio of an indefinitely small element of that arc to the radius; F=T1—T2 the total friction between the band and the pulley; dF the elementary portion of that friction due to the elementary arc dB; f the coefficient of friction between the materials of the band and pulley.
Then, according to a well-known principle in statics, the normal pressure at the elementary arc de is TdO, T being the mean tension of the band at that elementary arc; consequently the friction on that arc is dF=fTdo. Now that friction is also the difference
A
x
between the tensions of the band at the two ends of the elementary ~ temperature of one pound of pure water, at or near ordinary atmoarc, or dT =dF =fTdO; which equation, being integrated throughout
the entire arc of contact, gives the following formulae:
T,
hyp log. ,i.2 = fO
=
f°
T2 e
F = Ti — T2 = T1 (1—e—f9) = T2(efe — I)
When a belt connecting a pair of pulleys has the tensions of its two sides originally equal, the pulleys being at rest, and when the pulleys are next set in motion, so that one of them drives the other by means of the belt, it is found that the advancing side of the belt is exactly as much tightened as the returning side is slackened, so that the mean tension remains unchanged. Its value is given by this formula
2 2(ef°—1) (66)
which is useful in determining the original tension required to enable a belt to transmit a given force between two pulleys.
The equations 65 and 66 are applicable to a kind of brake called a friction-strap, used to stop or moderate the velocity of machines by being tightened round a pulley. The strap is usually of iron, and the pulley of hard wood.
Let a denote the arc of contact expressed in turns and fractions of a turn; then
eJ°=number whose common logarithm is 2.7288fa (67)
See also DYNAMOMETER for illustrations of the use of what are essentially friction-straps of different forms for the measurement of the brake horse-power of an engine or motor.
§ 104. Stiffness of Ropes.—Ropes offer a resistance to being bent, and, when bent, to being straightened again, which arises from the mutual friction of their fibres. It increases with the sectional area of the rope, and is inversely proportional to the radius of the curve into which it is bent.
The work lost in pulling a given length of rope over a pulley is found by multiplying the length of the rope in feet by its stiffness in pounds, that stiffness being the excess of the tension at the leading side of the rope above that at the following side, which is necessary to bend it into a curve fitting the pulley, and then to straighten it again.
The following empirical formulae for the stiffness of hempen ropes have been deduced by Morin from the experiments of Coulomb:—Let F be the stiffness in pounds avoirdupois; d the diameter of the rope in inches, n=48d2 for white ropes and 35d2 for tarred ropes; r the effective radius of the pulley in inches; T the tension in pounds. Then
For white ropes, F = r (0.0012+0•0o1o26n+o•oo12T For tarred ropes,F= r (0.006+0•0o1392n+o•oo168T
§ 105. Friction-Couplings.—Friction is useful as a means of communicating motion where sudden changes either of force or velocity take place, because, being limited in amount, it may be so adjusted as to limit the forces which strain the pieces of the mechanism within the bounds of safety. Amongst contrivances for effecting this object are friction-cones. A rotating shaft carries upon a cylindrical portion of its figure a wheel or pulley turning loosely on it, and consequently capable of remaining at rest when the shaft is in motion. This pulley has fixed to one side, and concentric with it, a short frustum of a hollow cone. At a small distance from the pulley the shaft carries a short frustum of a solid cone accurately turned to fit the hollow cone. This frustum is made always to turn along with the shaft by being fitted on a square portion of it, or by means of a rib and groove, or otherwise, but is capable of a slight longitudinal motion, so as to be pressed into, or withdrawn from, the hollow cone by means of a lever. When the cones are pressed together or engaged, their friction causes the pulley to rotate along with the shaft; when they are disengaged, the pulley is free to stand still. The angle made by the sides of the cones with the axis should not be less than the angle of repose. In the friction-clutch, a pulley loose on a shaft has a hoop or gland made to embrace it more or less tightly by means of a screw; this hoop has short projecting arms or ears. A fork or clutch rotates along with the shaft, and is capable of being moved longitudinally by a handle. When the clutch is moved towards the hoop, its arms catch those of the hoop, and cause the hoop to rotate and to communicate its rotation to the pulley by friction. There are many other contrivances of the same class, but the two just mentioned may serve for examples.
§ io6. Heat of Friction: Unguents.—The work lost in friction is employed in producing heat. This fact is very obvious, and has been known from a remote period; but the exact determination of the proportion of the work lost to the heat produced, and the experimental proof that that proportion is the same under all circumstances and with all materials, solid, liquid and gaseous, are comparatively recent achievements of J. P. Joule. The quantity of work which produces a British unit of heat (or so much heat as elevates thespheric temperatures, by I° F.) is 772 foot-pounds. This constant, now designated as " Joule's equivalent," is the principal experimental datum of the science of thermodynamics.
A more recent determination (Phil. Trans., 1897), by Osborne Reynolds and W. M. Moorby, gives 778 as the mean value of Joule's equivalent through the range of 32° to 212° F. See also the papers of Rowland in the Proc. Amer. Acad. (1879), and Griffiths, Phil. Trans. (1893).
The heat produced by friction, when moderate in amount, is useful in softening and liquefying thick unguents; but when excessive it is prejudicial, by decomposing the unguents, and sometimes even by softening the metal of the bearings, and raising their temperature so high as to set fire to neighbouring combustible matters.
Excessive heating is prevented by a constant and copious supply of a good unguent. The elevation of temperature produced by the friction of a journal is sometimes used as an experimental test of the quality of unguents. For modern methods of forced lubrication see BEARINGS.
§ 107. Rolling Resistance.—By the rolling of two surfaces over each other without sliding a resistance is caused which is called sometimes " rolling friction," but more correctly rolling resistance. It is of the nature of a couple, resisting rotation. Its moment is found by multiplying the normal pressure between the rolling surfaces by an arm, whose length depends on the nature of the rolling surfaces, and the work lost in a unit of time in overcoming it is the product of its moment by the angular velocity of the rolling surfaces relatively to each other. The following are approximate values of the arm in decimals of a foot:
Oak upon oak 0.006 (Coulomb).
Lignum vitae on oak 0.004
Cast iron on cast iron 0.002 (Tredgold).
§ Io8. Reciprocating Forces: Stored and Restored Energy.—When a force acts on a machine alternately as an effort and as a resistance, it may be called a reciprocating force. Of this kind is the weight of any piece in the mechanism whose centre of gravity alternately rises and falls; for during the rise of the centre of gravity that weight acts as a resistance, and energy is employed in lifting it to an amount expressed by the product of the weight into the vertical height of its rise; and during the fall of the centre of gravity the weight acts as an effort, and exerts in assisting to perform the work of the machine an amount of energy exactly equal to that which had previously been employed in lifting it. Thus that amount of energy is not lost, but has its operation deferred; and it is said to be stored when the weight is lifted, and restored when it falls.
In a machine of which each piece is to move with a uniform velocity, if the effort and the resistance be constant, the weight of each piece must be balanced on its axis, so that it may produce lateral pressure only, and not act as a reciprocating force. But if the effort and the resistance be alternately in excess, the uniformity of speed may still be preserved by so adjusting some moving weight in the mechanism that when the effort is in excess it may be lifted, and so balance and employ the excess of effort, and that when the resistance is in excess it may fall, and so balance and overcome the excess of resistance—thus storing the periodical excess of energy and restoring that energy to perform the periodical excess of work.
Other forces besides gravity may be used as reciprocating forces for storing and restoring energy—for example, the elasticity of a spring or of a mass of air.
In most of the delusive machines commonly called " perpetual motions," of which so many are patented in each year, and which are expected by their inventors to perform work without receiving energy, the fundamental fallacy consists in an expectation that some reciprocating force shall restore more energy than it has been the means of storing.
Division 2. Deflecting Forces.
§ 109. Deflecting Force for Translation in a Curved Path.—In machinery, deflecting force is supplied by the tenacity of some piece, such as a crank, which guides the deflected body in its curved path, and is unbalanced, being employed in producing deflexion, and not in balancing another force.
§ Ito. Centrifugal Force of a Rotating Body.-The centrifugal force exerted by a rotating body on its axis of rotation is the same in magnitude as if the mass of the body were concentrated at its centre of gravity, and acts in a plane passing through the axis of rotation and the centre of gravity of the body.
The particles of a rotating body exert centrifugal forces on each other, which strain the body, and tend to tear it asunder, but these forces balance each other, and do not affect the resultant centrifugal force exerted on the axis of rotation.'
If the axis of rotation traverses the centre of gravity of the body, the centrifugal force exerted on that axis is nothing.
Hence, unless there be some reason to the contrary, each piece of a machine should be balanced on its axis of rotation; otherwise the
' This is a particular case of a more general principle, that the motion of the centre of gravity of a body is not affected by the mutual actions of its parts.
(65)
T1+T2 ef9+1
B =6.2832a
(68)
centrifugal force will cause strains, vibration and increased friction, and a tendency of the shafts to jump out of their bearings.
§ I. Centrifugal Couples of a Rotating Body.—Besides the tend-
ency (if any) of the combined centrifugal forces of the particles of a rotating body to shift the axis of rotation, they may also tend to turn it out of its original direction. The latter tendency is called a centrifugal couple, and vanishes for rotation about a principal axis.
It is essential to the steady motion of every rapidly rotating piece in a machine that its axis of rotation should not merely traverse its centre of gravity, but should be a permanent axis; for otherwise the centrifugal couples will increase friction, produce oscillation of the shaft and tend to make it leave its bearings.
The principles of this and the preceding section are those which regulate the adjustment of the weight and position of the counter-poises which are placed between the spokes of the driving-wheels of locomotive engines.
§ I12.* Method of computing the position and magnitudes of balance weights which must be added to a given system of arbitrarily chosen rotating masses in order to make the common axis of rotation
a permanent axis.—The method here briefly explained is taken from a paper by W. E. Dalby, " The Balancing of Engines with special reference to Marine Work," Trans. Inst. Nay. Arch. (1899). Let the weight (fig. 130), attached to a truly turned disk, be rotated by the shaft OX, and conceive that the shaft is held
in a bearing at one point, O. The
force required to constrain the weight
to move in a circle, that is the de-
viating force, produces an equal and
opposite reaction on the shaft, whose X amount F is equal to the centrifugal
force Wa2r/g lb, where r is the radius
of the mass centre of the weight, and
a is its angular velocity in radians per
second. Transferring this force to
(From Balancing of Engines, by the point O, it is equivalent to, (I) permission of Edward Arnold.) a force at 0 equal and parallel to
F, and, (2) a centrifugal couple of Fa foot-pounds. In order that OX may be a permanent axis it is necessary that there should be a sufficient number of weights attached to the shaft and so distributed that when each is referred to the point 0
(I) IF =o
lFa =o (a)
The plane through 0 to which the shaft is perpendicular is called the reference plane, because all the transferred forces act in that plane at the point O. The plane through the radius of the weight containing the axis OX is called the axial plane because it contains the forces forming the couple due to the transference of F to the reference plane. Substituting the values of F in (a) the two conditions become
(1) (Wlri+W2rs+W2ri+...) a
8 =o
(2) (W1air1+W2a2r2+...) o g
In order that these conditions may obtain, the quantities in the brackets must be zero, since the factor a2/g is not zero. Hence finally the conditions which must be satisfied by the system of weights in order that the axis of rotation may be a permanent axis is
(I) (Wiri+W2r2+Wsra) =0 (c)
(2) (Wiair,+W2a2r2+) = 0
It must be remembered that these are all directed quantities, and that their respective sums are to be taken by drawing vector polygons. In drawing these polygons the magnitude of the vector of the type Wr is the product Wr, and the direction of the vector is from the shaft outwards towards the weight W, parallel to the radius r. For the vector representing a couple of the type War, if the masses are all on the same side of the reference plane, the direction of drawing is from the axis outwards; if the masses are some on one side of the reference plane and some on the other side, the direction of drawing is from the axis outwards towards the weight for all masses on the one side, and from the mass inwards towards the axis for all weights on the other side, drawing always parallel to the direction defined by the radius r. The magnitude of the vector is the product War. The conditions (c) may thus be expressed: first, that the sum of the vectors Wr must form a closed polygon, and, second, that the sum of the vectors War must form a closed polygon. The general problem in practice is, given a system of weights attached to a shaft, to find the respective weights and positions of two balance weights or counterpoises which must be added to the system in order to make the shaft a permanent axis, the planes in which the balance weights are to revolve also being given. To solve this the reference plane must be chosen so that it coincides with the plane of revolution of one of the as yet unknown balance weights. The balance weight in this plane has therefore no couple corresponding to it. Hence by drawing a couple polygon for the given weights the vector which is required to close the polygon is at once found and from it the magnitude and position of the balance weight which must be added to the system to balance the couples follow at once. Then, transferring the product Wr corresponding
with this balance weight to the reference plane, proceed to draw the force polygon. The vector required to close it will determine the second balance weight, the work may be checked by taking the reference plane to coincide with the plane of revolution of the second balance weight and then re-determining them, or by taking a reference plane anywhere and including the two balance weights trying if condition (c) is satisfied.
When a weight is reciprocated, the equal and opposite force required for its acceleration at any instant appears as an unbalanced force on the frame of the machine to which the weight belongs. In the particular case where the motion is of the kind known as " simple harmonic " the disturbing force on the frame due to the reciprocation of the weight is equal to the component of the centrifugal force in the line of stroke due to a weight equal to the reciprocated weight supposed concentrated at the crank pin. Using this principle the method of finding the balance weights to be added to a given system of reciprocating weights in order to produce a system of forces on the frame continuously in equilibrium is exactly the same as that just explained for a system of revolving weights, because for the purpose of finding the balance weights each reciprocating weight may be supposed attached to the crank pin which operates it, thus forming an equivalent revolving system. The balance weights found as part of the 'equivalent revolving system when reciprocated by their respective crank pins form the balance weights for the given reciprocating system. These conditions may be exactly realized by a system of weights reciprocated by slotted bars, the crank shaft driving the slotted bars rotating uniformly. In practice reciprocation is usually effected through a connecting rod, as in the case of steam engines. In balancing the mechanism of a steam engine it is often sufficiently accurate to consider the motion of the pistons as simple harmonic, and the effect on the framework of the acceleration of the connecting rod may be approximately allowed for by distributing the weight of the rod between the crank pin and the piston inversely as the centre of gravity of the rod divides the distance between the centre of the cross head pin and the centre of the crank pin. The moving parts of the engine are then divided into two complete and independent systems, namely, one system of revolving weights consisting of crank pins, crank arms, &c., attached to and revolving with the crank shaft, and a second system of reciprocating weights consisting of the pistons, cross-heads, &c., supposed to be moving each in its line of stroke with simple harmonic motion. The balance weights are to be separately calculated for each system, the one set being added to the crank shaft as revolving weights, and the second set being included with the reciprocating weights and operated by a properly placed crank on the crank shaft. Balance weights added in this way to a set of reciprocating weights are sometimes called bob-weights. In the case of locomotives the balance weights required to balance the pistons are added as revolving weights to the crank shaft system, and in fact are generally combined with the weights required to balance the revolving system so as to form one weight, the counterpoise referred to in the preceding section, which is seen between the spokes of the wheels of a locomotive. Although this method balances the pistons in the horizontal plane, and thus allows the pull of the engine on the train to be exerted without the variation due to the reciprocation of the pistons, yet the force balanced horizontally is introduced vertically and appears as a variation of pressure on the rail. In practice about two-thirds of the reciprocating weight is balanced in order to keep this variation of rail pressure within safe limits. The assumption that the pistons of an engine move with simple harmonic motion is increasingly erroneous as the ratio of the length of the crank r, to the length of the connecting rod 1 increases. A more accurate though still approximate expression for the force on the frame due to the acceleration of the piston whose weight is W is given by
g w2r cos 0 + l cos 20 i
The conditipns regulating the balancing of a system of weights reciprocating under the action of accelerating forces given by the above expression are investigated in a paper by Otto Schlick, " On Balancing of Steam Engines," Trans, Inst. Nay. Arch. (1900), and in a paper by W. E. Dalby, " On the Balancing of the Reciprocating Parts of Engines, including the Effect of the Connecting Rod " (ibid., 1901). A still more accurate expression than the above is obtained by expansion in a Fourier series, regarding which and its bearing on balancing engines see a paper by J. H. Macalpine, " A Solution of the Vibration Problem " (ibid., 1901). The whole subject is dealt with in a treatise, The Balancing of Engines, by W. E. Dalby (London, 1906). Most of the original papers on this subject of engine balancing are to be found in the Transactions of the Institution of Naval Architects.
§ 113.* Centrifugal Whirling of Shafts.—When a system of revolv-
ing masses is balanced so that the conditions of the preceding section are fulfilled, the centre of gravity of the system lies on the axis of revolution. If there is the slightest displacement of the centre of gravity of the system from the axis of revolution a force acts on the shaft tending to deflect it, and varies as the deflexion and as the square of the speed. If the shaft is therefore to revolve stably, this force must be balanced at any instant by the elastic resistance of the shaft to deflexion. To take a simple case, suppose a shaft,
(b)
supported on two bearings to carry a disk of weight W at its centre, and let the centre of gravity of the disk be at a distance e from the axis of rotation, this small distance being due to imperfections of material or faulty construction. Neglecting the mass of the shaft itself, when the shaft rotates with an angular velocity a, the centrifugal force Wa2e/g will act upon the shaft and cause its axis to deflect from the axis of rotation a distance, y say. The elastic resistance evoked by this deflexion is proportional to the deflexion, so that if c is a constant depending upon the form, material and method of support of the shaft, the following equality must hold if the shaft is to rotate stably at the stated speed
W (y+e)a2=cy,
g from which y=W a2e/(gc —We).
This expression shows that as a increases y increases until when Wa2=gc, y becomes infinitely large. The corresponding value of a, namely J gc/w, is called the critical velocity of the shaft, and is the speed at which the shaft ceases to rotate stably and at which centrifugal whirling begins. The general problem is to find the value of a corresponding to all kinds of loadings on shafts supported in any manner. The question was investigated by Rankine in an article in the Engineer (April 9, 1869). Professor A. G. Greenhill treated the problem of the centrifugal whirling of an unloaded shaft with different supporting conditions in a paper " On the Strength of Shafting exposed both to torsion and to end thrust," Proc. Inst. Mech. Eng. (1883). Professor S. Dunkerley (" On the Whirling and Vibration of Shafts," Phil. Trans., 1894) investigated the question for the cases of loaded and unloaded shafts, and, owing to the complication arising from the application of the general theory to the cases of loaded shafts, devised empirical formulae for the critical speeds of shafts loaded with heavy pulleys, based generally upon the following assumption, which is stated for the case of a shaft carrying one pulley: If Ni, N2 be the separate speeds of whirl of the shaft and pulley on the assumption that the effect of one is neglected when that of the other is under consideration, then the resulting speed of whirl due to both causes combined may be taken to be of the form N,N2J (N2I+N22) where N means revolutions per minute. This form is extended to include the cases of several pulleys on the same shaft. The interesting and important part of the investigation is that a number of experiments were made on small shafts arranged in different ways and loaded in different ways, and the speed at which whirling actually occurred was compared with the speed calculated from formulae of the general type indicated above. The agreement between the observed and calculated values of the critical speeds was in most cases quite remarkable. In a paper by Dr C. (Three, " The Whirling and Transverse Vibrations of Rotating Shafts," Proc. Phys. Soc. Lon., vol. 19 (1904); also Phil. Mag., vol. 7 (1904), the question is investigated from a new mathematical point of view, and expressions for the whirling of loaded shafts are obtained without the necessity of any assumption of the kind stated above. An elementary presentation of the problem from a practical point of view will be found in Steam Turbines, by Dr A. Stodola (London, 1905).
§ 114. Revolving Pendulum. Governors.—In fig. 131 AO represents an upright axis or spindle; 13 a weight called a bob, suspended by rod
OB from a horizontal axis at 0, carried
E by the vertical axis. When the spindle is at rest the bob hangs close to it; when the spindle rotates, the bob, being made to revolve round it, diverges until the resultant of the centrifugal force and the weight of the bob is a force acting at O in the direction OB, and then it revolves steadily in a circle. This combination is called a revolving, centrifugal, or conical pendulum. Revolving pendulums are usually constructed with pairs of rods and bobs, as OB, Ob, hung at opposite sides of the spindle, that the centrifugal forces exerted at the point 0 may balance each other.
In finding the position in which the bob will revolve with a given angular velocity, a, for most practical cases connected with machinery the mass of the rod may be considered as insensible compared with that of the bob. Let the bob be a sphere, and from the centre of that sphere draw BH =y perpendicular to OA. Let OH = z; let W be the weight of the bob, F its centrifugal force. Then the condition of its steady revolution is W: F::z: y; that is to say, y/z=F/W =ya2/g; consequently
z = g/a2 (69)
Or, if n = a;'27r = a/6.2832 be the number of turns or fractions of a turn in a second,
g 0.8165 ft. _9.79771 in.
47r2.n2 — n2 — n2
s is called the altitude of the pendulum.
If the rod of a revolving pendulum be jointed, as in fig. 132, not to a point in the vertical axis, but to the end of a projecting arm C, the position in which the bob will revolve will be the same as if the rod were jointed to the point 0, where its prolongation cuts the vertical axis.
A revolving pendulum is an essential part of most of the contrivances called governors, for regulating the speed of prime movers, for further particulars of which see STEAM